The locus of the feet of the perpendiculars drawn from either focus onto a variable tangent to the hyperbola $16y^2 - 9x^2 = 1$ is

If $c$ is a real number and $\frac{x^2}{c-12}+\frac{y^2}{7-c}=1$ represents a hyperbola,then

If $(1,2)$ is the focus,$x+2y=0$ is the directrix,and $\sqrt{2}$ is the eccentricity of a hyperbola,then the equation of the hyperbola is

The foci of a hyperbola are $(\pm 2, 0)$ and its eccentricity is $\frac{3}{2}$. $A$ tangent,perpendicular to the line $2x + 3y = 6$,is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the $x$- and $y$-axes are $a$ and $b$ respectively,then $|6a| + |5b|$ is equal to $..........$.

Let the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ be reciprocal to that of the ellipse $x^{2}+9y^{2}=9$. Then the ratio $a^{2}:b^{2}$ equals:

If $\theta$ is the acute angle between the tangents drawn from the point $(2,3)$ to the hyperbola $5x^2-6y^2-30=0$,then $\tan \theta=$